Langmuir 2006, 22, 2975-2978
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Ion Adsorption and Lamellar-Lamellar Transitions in Charged Bilayer Systems Jan Forsman Theoretical Chemistry, Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden ReceiVed January 5, 2006. In Final Form: February 9, 2006 Using a primitive model approach, we analyze the influence of ion specific adsorption on the phase behavior of charged lamellar systems. The presence of a weak short-ranged surface potential, attracting monovalent counterions, induces a phase separation, where the separate phases have different repeat distance. If the adsorption potential is very weak, the more narrow phase never forms. An opposite behavior is found for strong surface affinities. Both Monte Carlo simulations and a recently developed correlation-corrected Poisson-Boltzmann theory are adopted, with a nearly quantitative agreement between the approaches. Different counterions are discriminated by the adsorption potential strength, and with physically reasonable values, experimental observations on these systems are well reproduced. The study highlights the importance of electrostatic correlations, even though only monovalent ions are present.
1. Introduction Interactions between charged surfaces have been extensively studied for many decades, with the DLVO theory1,2 as the fundamental cornerstone. Usually, the solvent is approximated as a dielectric continuum: the primitive model. In many cases, interactions arising from the distribution of ions are approximated by the Poisson-Boltzmann (PB) theory. PB calculations are simple, fast, noise-free, and often reasonably accurate in systems with weak electrostatic coupling. Its major flaw is the inability3 to even qualitatively reproduce attractions due to ion-ion correlations. Such attractions are known to exist in highly coupled systems,4,5,6 e.g., in the presence of multivalent ions. However, as we shall see below, correlation attractions can sometimes be relevant in relatively weakly coupled systems. To capture such effects, we shall adopt a recently proposed correlation-corrected version of the PB theory:7 cPB for short. This theory attempts to account for ion-ion correlations in an effective manner, while retaining the mean-field character of the mother theory. It will be briefly described below, while a detailed account may be found in ref 7. In aqueous solution, the cationic double-chain surfactant DDABr (didodecyldimethylammonium bromide) is known to display a phase separated regime, in which two lamellar forms coexist.8-11 These are generally denoted LR and LR′, where the former is “swollen”, with a relatively large repeat distance, whereas the latter, “collapsed”, form displays a very narrow aqueous slit. Hence, if the osmotic pressure is increased isothermally from a low value, the lamellar spacing will initially decrease monotonically, but as the coexistence pressure is exceeded, there will be a sudden jump to the collapsed state. Interestingly, this behavior is sensitive to the properties of the (1) Derjaguin, B. V.; Landau, L. Acta Phys. Chim. URSS 1941, 14, 633-662. (2) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier Publishing Company Inc.: Amsterdam, 1948. (3) Sader, J. E.; Chen, D. Y. C. Langmuir 2000, 16, 324. (4) Oosawa, F. Polyelectrolytes; Marcel Dekker: New York, 1971. (5) Patey, G. N. J. Chem. Phys. 1980, 72, 5763. (6) Guldbrand, L.; Jo¨nsson, B.; Wennerstro¨m, H.; Linse, P. J. Chem. Phys. 1984, 80, 2221. (7) Forsman, J. J. Phys. Chem. B 2004, 108, 9236. (8) Warr, G. G.; Sen, R.; Evans, F. J. Phys. Chem. 1988, 92, 774. (9) Dubois, M.; Zemb, T. Langmuir 1991, 7, 1352. (10) Zemb, T.; Belloni, L.; Dubois, M.; Marcelja, S. Prog. Colloid Polym. Sci. 1992, 89, 33. (11) Dubois, M.; Zemb, T.; Fuller, N.; Rand, R. P.; Parsegian, V. A. J. Chem. Phys. 1998, 108, 7855.
surfactant counterion. If bromide ions (Br-) are replaced by chloride ions (Cl-), the LR′ phase is absent, and the lamellar repeat spacing decreases monotonically as the osmotic pressure is raised.11,12 Ion specific effects in general, and surface adsorption in particular, have been the focus of many experimental and theoretical studies in recent years.13-18 The consensus today is that halide ions are attracted to surfaces, or at least to the airwater interface. Furthermore, the excess adsorption at the interface, Γ, increases with size and polarizability, i.e., Γ(Cl-) < Γ(Br-) < Γ(I-). It seems natural to assume that there is a close connection between these properties and the ion specific effects observed in the lamellar surfactant systems. In this work, we will pursue this conjecture and try to rationalize the observed phase behavior and osmotic pressure response, in terms of ion specific surface affinities.
2. Theoretical Model We shall conduct our theoretical analyses using the restricted primitive model. Specifically, we focus on the aqueous region confined between adjacent layers of charged amphiphiles. These are modeled as two infinite planar and parallel walls, carrying a uniform surface charge density of σ ) e/70 Å-2, where e is the elementary charge. Dissolved ions are assumed to be pointlike and occupy the region between the walls, but do also exchange with a bulk solution in cases where the salt concentration is finite. A uniform dielectric response of r ) 78.3 is assumed, throughout the system. Excess ion adsorption is introduced via an external potential, w(z), emanating from each surface in the z direction normal to the surfaces. This potential may be regarded as a potential of mean force, effectively including solvent-induced effects in the vicinity of such surfaces. Its exact appearance is of course not known, but guided by simulation results at waterair interfaces,18 we expect it to be relatively weak and have a (12) Kang, C.; Khan, A. J. Colloid Interface Sci. 1993, 156, 218. (13) Stuart, S. J.; Berne, B. J. J. Phys. Chem. A 1999, 103, 10300. (14) Ghosal, S.; Shbeeb, A.; Hemminger, J. C. Geophys. Res. Lett. 2000, 27, 1879. (15) Knipping, E. M.; Lakin, M. J.; Foster, K. L.; Jungwirth, P.; Tobias, D. J.; Gerber, R. B.; Dabdub, D.; Finlayson-Pitts, B. J. Science 2000, 288, 301. (16) Jungwirth, P.; Tobias, D. J. J. Phys. Chem. B 2002, 106, 6361. (17) Ghosal, S.; Hemminger, J. C.; Bluhm, H.; Mun, B. S.; Hebenstreit, E. D. L.; Ketteler, G.; Ogletree, D. F.; Requejo, F. G.; Salmeron, M. Science 2005, 307, 563. (18) Mucha, M.; Frigato, T.; Levering, L. M.; Allen, H. C.; Tobias, D. J.; Dang, L. X.; Jungwirth, P. J. Phys. Chem. B 2005, 109, 7617.
10.1021/la0600393 CCC: $33.50 © 2006 American Chemical Society Published on Web 03/07/2006
2976 Langmuir, Vol. 22, No. 7, 2006
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Figure 1. Examples of typical ion adsorption potentials used in this study. The decay parameter τ is fixed at 2 Å, while the amplitude factor Aw is an adjustable parameter, with which the minimum of the adsorption potential, wmin, is regulated.
range comparable to the size of a water molecule. We shall discuss this further below, but for now we merely state our choice for the functional appearance of w(z):
βw(z) ) Awe-z/τΦL-J(z)
(1)
where the wall is located at z ) 0 and β ) 1/(kT) is the inverse thermal energy. Aw and τ are parameters that regulate the strength and, to some extent, the range of the adsorption potential. ΦL-J is the potential from a semi-infinite half-space of close-packed Lennard-Jones particles: βΦL-J(z) ) 2π[2/45(σw/z)9 1/ (σ /z)3], where we have set σ ) 4 Å. The adsorption potential 3 w w from each wall is assumed to be additive; that is, the net surface interaction of an ion at a position z, when the surface separation is h, is given by W(z, h) ) w(z) + w(h - z). The influence of τ on the observed surface forces is rather limited, since ΦL-J decays relatively fast even with an infinite value of τ. Hence, we have chosen to use a fixed value: τ ) 2 Å, ensuring that the adsorption potentials are short-ranged. This leaves the amplitude factor, Aw, with which we regulate the surface affinity of the surfactant counterions. Examples of surface potentials, relevant to this study, are provided in Figure 1. We have chosen to quote the minimum value of w(z), wmin, rather than Aw itself. This is because the “depth” of the adsorption well is more intuitive and has a more direct relation to, for instance, experimentally observed or simulated ion distributions near surfaces. It is clear from the graph that the typical adsorption potentials we invoke are relatively weak and short-ranged. The region in which βw(z) < -0.1 only covers a distance of about 3-5.5 Å from the surface, which should serve as an estimate of the effective range. Judging from recent simulation results by Jungwirth and co-workers,16,18 on aqueous ion solutions near an air-water interface, such a range is quite reasonable. Furthermore, typical values of maximum excess ion concentration ratios (relative to the bulk value) in those simulations are 1.5-5. This implies a maximum attraction well for a corresponding potential of mean force of -ln[1.5]kT ≈ - 0.4kT (for Cl-), up to -ln[5]kT ≈ - 1.6kT (for iodide, I-), suggesting that our choices of typical potential depths amount to representative values. However, these values can only serve as a rough guide, since the simulations were made at water-air interfaces, not at water-surfactant layer interfaces. We stress that the results we obtain are qualitatively insensitive to the functional form of the surface potential. Similar pressure curves, and predicted phase diagrams, are obtained with pure Lennard-Jones surfaces (implying van der Waals interactions as the source of ion specificity). Most of our analyses will be made with the recently proposed correlation-corrected Poisson-Boltzmann theory,7 cPB, but we
also compare with corresponding simulation results and predictions by standard Poisson-Boltzmann theory, PB. In the cPB theory, effects of anti-correlations between ions of like charge, which are neglected in the ordinary PB approach, are effectively included by a change of the Hamiltonian at short ion-ion separations. In other words, the mean-field character of the PB theory is retained, but the artifacts of a vanishing radial distribution function at short separations (for ions of like charge) are effectively compensated by a reduction of the repulsive interaction at short range. If we, for the sake of convenience, restrict ourselves to the description of neutralizing monovalent counterions in a salt free system, we can write down a free energy functional, F [n(r)], as
βF [n(r)] )
∫n(r)(ln n(r) - 1) dr +
∫∫n(r) n(r′) φ(|r - r′|) dr′ dr + β∫Vext(r) n(r) dr
β 2
(2)
where n(r) is the counterion density at position r, φ(|r - r′|) is the interaction potential between counterions and Vext(r) is an external potential, which in our case originates from the uniform surface charge as well as from the additional adsorption potential W(z, h). The extension to systems containing salt and multivalent ions is straightforward7 but complicates the notation. In the ordinary PB theory, φ(r) is simply the Coulomb potential: φPB(r) ) φCoul(r) ) lB/r, where lB ) βe2/(4π0r) is the Bjerrum length. The relative permittivity of vacuum is denoted 0. In the cPB approach, this mean-field interaction potential between ions of like charge is reduced at separations below a threshold value Rc
φcPB(r) )
[
φCoul(r), r > Rc φcorr(r), r e Rc
]
(3)
where
φcorr(r) )
( ) dφCoul dr′
(r - Rc) + φCoul(Rc)
r′)Rc
(4)
With this choice, both φcPB and its derivative are continuous across the transition distance Rc. Obviously, Rc should reflect the separation below which ion correlations become important. Forsman7 suggested the following expression, for monovalent ions in the presence of a uniformly charged surface:
Rc )
xπ|σ|e
(5)
This relation can be motivated a follows: imagine that counterions condense at the surface to the extent that it becomes perfectly neutralized. Assuming a circular area per ion, Rc is the radius of this circle. Hence, we approximate the transition distance from the proximity between counterions, as they crowd at the surface. Although this is a reasonable estimate, it is still largely motivated by the accuracy obtained, even in highly coupled systems. Notice that the precise way in which the correlationcorrection to the mean field is formulated is not important. Even a simple “Coulomb hole” will lead to a considerable improvement, i.e., to a mean-field theory fully able to handle known effects of ion-ion correlations, such as attractions between aggregates of equal charge and charge reversal. Still, the above choice, with a “soft” transition, has some extra physical appeal. It has also been shown to generate remarkably accurate predictions, even in strongly coupled systems, and at high as well as low salt concentrations.
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Langmuir, Vol. 22, No. 7, 2006 2977
The grand potential, Ω, is obtained from βΩ ) βF + 2πlBσ2h/ Systems containing salt require an additional term: -∑RµR∫0hnR(x′) dx′, where µR is the chemical potential of ion R. The osmotic pressure, Π, is given by Π ) S-1δΩeq/δh, where Ωeq is the equilibrium value of the grand potential and S is the surface area. The osmotic pressure can be calculated, either as a discrete derivative of Ωeq with respect to the separation, or directly via the force exerted between ions and walls as well as between the charged walls themselves. Details on grand potential minimization, etc., are provided in ref 7. Results from canonical Metropolis Monte Carlo simulations will be denoted “MC”. Standard periodic boundary conditions were employed in the directions parallel with the surfaces. A long-range correction was added, to account for interactions with charges outside the simulation box.19 Checks were performed to ensure that size effects can be neglected. The temperature is always 298 K in this work. Using canonical simulations implies bulk conditions with a vanishing amount of salt. Effects of salt addition will be investigated via cPB calculations. e2.
3. Results and Discussion In Figure 2, we compare simulated data on how the osmotic pressure varies with the surface separation in salt-free systems, with corresponding predictions by the cPB and PB theories, for various adsorption potential strengths. The agreement between cPB and MC is almost quantitative in all cases, with a clearly discernible van der Waals loop in the osmotic pressure curves, implying phase separation. The width of the aqueous regions for two coexisting phases, can be established by a Maxwell construction.20,21 For example, with βwmin ) -0.75, cPB predicts equilibrium between phases in which the aqueous layer thicknesses are 7.3 and 23 Å, respectively. The corresponding values obtained from simulations are 7.3 and 25 Å, whereas ordinary PB calculations generate conditions very close to critical; that is, LR and LR′ have almost merged together. Note that when ion-ion correlations are taken into account even a modest adsorption potential will induce phase separation. The PB theory requires much stronger surface affinities in order to predict this phenomenon. An important difference between the approaches is that PB theory never will predict an attractive osmotic pressure unless there is a significant overlap between w(z) and w(h - z). Hence, should we adopt a surface potential with a strictly limited range, PB will not predict any attraction at separations where the central portion of the slit is outside the range of the surface potential. This is, however, not the case for cPB predictions or simulated results, where a net attraction still is present, provided the adsorption potential is strong enough. This attraction is due to electrostatic correlations between ions on either side of the mid plane of the slit. Having confirmed the accuracy of the cPB theory, we shall now investigate in more detail how the phase behavior of the system changes with the strength of the adsorption potential. This amounts to a shift of the surface affinity for the counterions in the corresponding physical system. In Figure 3a, we see how the two-phase regime gradually disappears as the depth of the adsorption potential diminishes. Comparing with experiments on DDACl, where the LR phase is stable at all pressures, we would conclude that in this model system chloride ions are best described by βwmin > -0.3. (19) Torrie, G. M.; Valleau, J. P. Chem. Phys. Lett. 1979, 65, 343. (20) Khan, A.; Jo¨nsson, B.; Wennerstro¨m, H. J. Phys. Chem. 1985, 89, 5180. (21) Turesson, M.; Forsman, J.; Åkesson, T.; Jo¨nsson, B. Langmuir 2004, 20, 5123.
Figure 2. Comparisons between predictions by the PB and cPB theories with simulation results. The osmotic pressure acting between the charged surfaces is denoted Π, while h is the surface separation. Several curves, with different choices of the adsorption potential depth, are given. In the physical system we wish to mimic, each curve corresponds to a system in which the counterions have a certain surface affinity.
Comparisons with independent simulation data16,18 suggest that this is a reasonable number. In an analogous manner, the phase behavior of DDABr is quite well reproduced if we set βwmin ) -1, which again conforms reasonably well with simulation results and experimental data16,17,18 of ion accumulation at the airwater interface. Using values of βwmin ) -0.25 for DDACl and βwmin ) -1 for DDABr, the graph in Figure 3b illustrates the predicted osmotic pressure curves for these systems. The predicted coexistence pressure for DDABr is about 0.4 MPa, and the aqueous layer thicknesses of the coexisting phases are 7.1 and 48 Å, respectively.22 This is in excellent agreement with experimental data on DDABr, and a similar conclusion applies for the model of DDACl.11 The agreement we observe is remarkable, escpecially considering that the aqueous layer of the LR′ is narrow; that is, one would expect the primitive model to provide a very crude description of the system. We also note in Figure 3a that with counterions adsorbing strongly enough the LR′ phase will always be more stable than (22) cPB calculations of the same system, but with an adsorption potential given by the full Lennard-Jones potential, with βwmin ) 0.87, results in a pressure curve very similar to the one obtained for DDABr.
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Figure 4. cPB predictions on the dependence of how the LR - LR′ phase equilibrium is affected by the addition of salt, for a system with βwmin ) -1. This system is our model representation of DDABr.
Figure 3. (a) cPB predictions of how the LR - LR′ phase equilibrium depends on the properties of the counterion, i.e., on the depth of the adsorption potential. The bulk solution contains no salt: cs ) 0. (b) cPB predictions of osmotic pressure curves, in salt free systems, for models of DDACl (βwmin ) -0.25) and DDABr (βwmin ) -1). In the latter case, the phase transition between LR′ and LR leads to a plateau regime.
LR. This means that our analysis predicts that swelling will not occur in a system like DDAI (disregarding phases different from LR and LR′). Unfortunately, we are not aware of any experimental measurements on DDAI that can verify, or disprove, this prediction. Let us finally discuss what happens if we couple our system to a reservoir containing a monovalent salt. We will specifically investigate our model of DDABr, i.e., βwmin ) -1. The predicted effect of salt addition is summarized in Figure 4. Interestingly, the initial response is weak. However, at higher bulk salt concentration, adding more salt will generate a dramatic increase of the aqueous layer thickness of the LR phase. At high enough salt concentrations, only LR′ is stable, and the system will not swell. The abrupt salt dependence sets in when screening starts to significantly suppress the repulsive regime of the van der Waals loop, which finally prevents a proper Maxwell construction; that is, the attractive regime cannot be compensated
by a corresponding repulsive one. Systems containing salt have not been simulated in this study, but previous comparisons have confirmed that the cPB theory accurately predicts salt effects, even at much higher concentrations and in the presence of multivalent species.7 Such systems display effects (packing, strong correlations, etc.) that are considerably more difficult to capture than in the present system, so we are confident that the cPB predictions are semiquantitatively correct.
4. Conclusions We have found that the phase behavior and osmotic pressure curves observed in concentrated solutions containing DDABr and DDACl, respectively, can be rationalized with a primitive model description in which the halide ions display an excess affinity to the surfaces of the surfactant layers. The affinity is, in accordance with recent experimental and theoretical evidence, assumed to be stronger for bromide ions. Thus, although a primitive model description is adopted, some solvent driven effects are implicitly included via an approximate ion-surface potential of mean force. The stronger surface affinity of the bromide ions leads to the formation of two separate lamellar phases, with different repeat spacings. Predictions by a recently formulated correlation-corrected version of the PoissonBoltzmann theory are in almost quantitative agreement with corresponding simulation results, whereas the original PB theory performs rather poorly. This demonstrates the importance of ion-ion correlations in these systems, which may be somewhat surprising, given the absence of multivalent species. LA0600393
